Why is e^(ln x)=x? (O Level Math/ A Level Math Tuition)

Why is \boxed{e^{\ln x}=x}?

This formula will be useful for some questions in O Level Additional Maths, or A Level H2 Maths.

There are two ways to show or prove this, first we can let

y=e^{\ln x}

Taking natural logarithm (ln) on both sides, we get

\ln y=\ln x\ln e=\ln x

So y=x. Substitute the very first equation and we get e^{\ln x}=x. 🙂

Alternatively, we can view e^x and \ln x as inverse functions of each other. So, we can let f(x)=e^x and f^{-1}(x)=\ln x. Then, e^{\ln x}=f(f^{-1})(x)=x by definition of inverse functions. This may be a better way to remember the result. 🙂

The above method of inverse functions can be used to remember \ln (e^x)=x too.

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